A Studentized Spherical Harmonics–Based Nonparametric Two-Sample Test for Compositional and Directional Data

Compositional data analysis has gained increasing attention due to the widespread occurrence of simplex-valued data, including microbiome data. However, existing kernel or distance-based nonparametric two-sample tests are often designed for Euclidean data and rely on square-root or log-transformations, motivating the need for a unified framework for nonparametric two-sample testing applicable to both compositional and directional data. We propose a studentized spherical harmonic energy distance-based two-sample test over a fixed dimensional underlying space, incorporating U-statistics theory and recent developments of studentization in the context of compositional and directional data. We establish asymptotic normality of our studentized test statistics constructed via spherical harmonics theory, avoiding the need for permutation or bootstrap tests. Simulations demonstrate convergence to the limiting distribution, empirical size control, and improved power in certain scenarios. Our proposed framework paves a new direction for nonparametric testing in non-Euclidean data analysis.